Parallelograms
(B D
and
E G)
on equal bases and between the same parallels are equal.

Draw the right lines
B F
and
C G.

Because the lines
B C
and
F G
are equal to the same
E H(XXXIV),
they are equal to one another;
ABCDFEHG
but they are also parallel, therefore
B F
and
C G
which join their extremities are parallel
(XXXIII),
and
B G
is a parallelogram; therefore equal to both
B D
and
E G(XXXV),
and therefore the parallelograms
B D
and
E G
are equal.

It is here supposed that the equal bases are placed in the
same right line.

(167)Cor.—If
two opposite sides of a parallelogram be divided into the
same number of equal parts, and the corresponding points of
division be joined by right lines, these right lines will
severally divide the parallelogram into as many equal
parallelograms.